Hello again. I suspect you no longer have the branches. I'll go ahead and answer as best I can without being able to see them.
The reason you haven't found any information on borers in burning bush is that they would be a very unusual event in that species. But that doesn't mean it's not possible. If you're fairly sure that's the problem, you can take measures to try to help.
Adult borers are actually moths that fly in the spring. They lay eggs under the bark, and the larvae then burrow their way through the shrub. It can take up to two years to see the damage. It is typical for heavily damaged branches to be black. In addition, borers open up a tree to fungal infections, which do further damage.
The damage to your bush is already done and can't be undone, since some branches are dead. You can still attempt to save it. It may or not be possible, depending on the extent of the damage.
If you want to try to do this on your own, the first step would be to cut off all the dead wood and destroy it. You have already done this. If you want to use a spray, you have to spray for borers in the spring when the adults are flying. Chlorpyrifos is sprayed twice, around mid-April and again in early May. This will have to be repeated each year. Chlorpyrifos is available in nurseries and garden centers.
To eliminate any fungus already introduced to the shrub, you can use a product called Agri-fos. It is available in garden centers, on Amazon.com, and many other online sources. It is also mixed with water and poured in as a soil drench, now and again next spring.
If you have more questions, let me know. If your bush are not too heavily damaged, these methods may save it. I hope that will be the case.
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